Question

Determine whether the pair of lines represented by the equations are parallel, perpendicular, or neither. $$ x-3=0 \quad \text { and } \quad y-5=0 $$

Short answer

The given equations are \(x = 3\) and \(y = 5\). The first line is a vertical line with an undefined slope, and the second line is a horizontal line with a slope of 0. Although their slopes don't follow the rule for perpendicular lines, one line is vertical and the other is horizontal, forming a right angle. Therefore, they are perpendicular.

Step-by-step solution

Identify the equations of the lines

The given equations are \(x - 3 = 0\) and \(y - 5 = 0\).

Rewrite the equations in slope-intercept form

The slope-intercept form of a linear equation is \(y = mx + b\), where m is the slope and b is the y-intercept. Let's rewrite the given equations in slope-intercept form: 1. For the equation \(x - 3 = 0\), we can rewrite it as \(x = 3\). Since there is no y term, this is a vertical line. 2. For the equation \(y - 5 = 0\), we can rewrite it as \(y = 5\). Since there is no x term, this line is horizontal.

Calculate the slopes of the two lines

Now that the equations are in slope-intercept form, we can determine the slopes of the two lines: 1. For the vertical line \(x=3\), there is no slope because vertical lines have undefined slopes. 2. For the horizontal line \(y=5\), the slope is zero, because horizontal lines have a slope of 0.

Determine the relationship between the lines

Since the first line has an undefined slope and the second line has a slope of 0, these lines do not have equal slopes or negatives reciprocals of their slopes. Therefore, these lines are neither parallel nor perpendicular. However, we can observe that one line is horizontal, and the other line is vertical, meaning that they form a right angle and are actually perpendicular.